TLM for diffusion: consistent first time step. Two‐dimensional case

Abstract

SUMMARYIn initializing a transmission line matrix (TLM) diffusion model it is necessary to consider both initial concentration (temperature) and initial flow. As usual, only one of them is given; an auxiliary formula is necessary to calculate the distribution for the first time step. It has been shown that the standard formula may introduce additional numerical errors (International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 1993; 6:135; International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 1993; 6:161) and that these errors can persist over many time steps. In this paper, we show how an initial modification to the normal TLM algorithm can remove such errors, and we demonstrate the method by applying it to a two-dimensional TLM heat diffusion model for a copper plate.

TLM nodal state estimator: An alternative method of initializing a two‐dimensional diffusion model

“…Comparing the results against Enders' similar error plots [10], the estimation method presented a faster convergence with the reference model. Despite a larger initial error in the first few timesteps of the estimation method compared with Ender's method, this error rapidly decreases to a smaller value in the long run.…”

“…Hence, when thermal problems involve prescribed boundary and stimulation temperature conditions, implementation of a TLM estimator can be used to derive the corresponding heat flow pulses which form an inherent part of the boundary condition. This in turn should improve model representation of the early portion of thermal transients which have been shown to be inaccurate at times [10].…”

“…The authors are indebted to David M. Stubbs for helpful insight into the implementation of Peter Ender's first timestep paper [10].…”

“…Enders highlighted the problem of model initialization in a two-dimensional TLM diffusion situation [10]; if a boundary or start-up condition is specified in terms of nodal temperature then the local pulse distribution must correspond to that temperature if the condition is to be well represented. The proposed new first timestep method was tested against the transient decay of a Gaussian temperature profile centred within a 10 Â 10 cm 2 plate.…”

“…The comparisons of the analytical reference and estimation results were made at the central node ð50; 50Þ: Figure 8 illustrates the analytical and estimated transient response at the centre of the plate. To compare Enders' modified first timestep method with the use of estimation for initialization, the relative percentage error between the estimation and analytical solutions at any point in space and time are calculated using [10] 100 Â T numericalðmDx;nDy;kDtÞ T analyticalðx;y;tÞ À 1 % ð31Þ…”